In further numerical experiments, we have observed that the lack of absolute monotonicity strongly effectively affects the computation of density and, as a consequence, the stability of the whole numerical scheme. For Courant number around \(C\approx 0.2\)  the original method becomes unstable, while the modified scheme with \(\alpha = 0.5\)  is still able to recover the expected convergence rates at least in the \(k=1 \)  case, as evident from Table REF , while again in the \(k=2\)  reported in Table REF  we observe a degradation of the convergence rates. In order to be able to run at slightly longer time steps we have then chosen
to use the \( \alpha = 0.5 \)  value for the IMEX scheme for the rest of the numerical simulations carried out in this paper.
We notice also that, for both schemes, the results compare
well with the analogous results presented in [1]} and with those obtained in [2]} with a higher order
IMEX method.
<TABLE><TABLE>